Shear Correction Factor Effects on Functionally Graded Materials (FGMs) Beams using Discrete Shear Gap (DSG) Element

Materials and computational methods are essential to support the development of infrastructure. Composite material has been used in many applications; in the laminated composite, failure due to excessive interlaminar stresses between two materials causes delamination. Thus, functionally graded materials (FGMs) have emerged. A numerical computation such as the finite element method (FEM) is widely used to support the analysis of FGMs in structural applications. The discrete shear gap DSG element is developed using Timoshenko beam theory, where the shear correction factor is used in their formulation. The shear correction factor is assumed to be constant in many applications; thus, it is valid for isotropic homogenous material. However, the effect of shear deformation significantly impacts the results of the FGMs beam, so the shear correction factor cannot be considered constant. Therefore, this paper presents the shear correction factor effect on static analysis of FGMs beam using DSG element. Various boundary conditions with length thickness ratio (L/h = 4) are evaluated. The DSG element yields good results in FGMs beam for different power-law index ratios. Furthermore, the DSG element result shows that the higher the modulus of elasticity ratio of the top-to-bottom material, the further the difference between k FGMs and k = 5/6 (constant). The DSG element can provide precise results without shear locking.

Composite; DSG; FEM; FGMs; Shear correction factors

Composites are the combination of materials selected based on the variety of the physical properties of each constituent material. They are combined to produce a new material with unique properties compared to those of the primary material. The application of composite material in beam structures can be found in Benbouras et al. (2017).  Delamination remains a large problem in laminated composites (Reddy, 2006). In an attempt to solve this problem, new composite materials, called Functionally Graded Materials (FGMs), were first proposed in 1984 by a group of materials scientists in Japan. 

The material properties vary gradually and continuously from one surface to another according to the given function. The gradation of material properties through thickness avoids sudden changes in stress distribution. In general, this material is a mixture of ceramics on the top and metal at the bottom. The main benefit of using this material is that it can withstand extreme situations like a high-temperature environment. The use of this material increases rapidly in many applications. Numerical analysis such as FEM is desired to support the development of FGMs.

 FEM is one of the numerical techniques used to support structural analysis. Many papers deal with the recent development of FEM in plate and shell structures for isotropic material (Katili et al., 2014; Maknun et al., 2016; Irpanni et al., 2017; Katili et al., 2017; Katili et al., 2018a; Katili et al., 2018b; Katili et al., 2019a). The results of FEM for the composite material can be found in (Katili et al., 2015; Maknun et al., 2015; Katili et al., 2018c; Katili et al., 2018d; Katili et al., 2019b; Maknun et al., 2020). From the results, FEM can support the analysis of plate and shell structures in isotropic and composite materials with accurate results.

 FEM performs very well in calculating FGMs beam (Simsek, 2009; Thai and Vo 2012; Nguyen et al., 2013, Aghazadeh et al., 2014; Jing et al., 2016; Katili and Katili 2020). Many beam elements have been proposed for application in beam structures using either Euler Bernoulli's or Timoshenko’s (Timoshenko, 1921; Timoshenko, 1922) beam theories. The beam element developed using Euler Bernoulli's beam theory needs C¹ continuity and can only be used for thin beam structures. Timoshenko beam elements were proposed to overcome these limitations. Beam elements developed from Timoshenko beam theory can be used for thin to thick beam structures and need only C⁰ continuity. However, the shear locking phenomenon exists in Timoshenko beam elements. There are several methods to overcome this phenomenon.  One of these is the discrete shear gap (DSG) method proposed by Bletzinger et al. (2000). 

For the beam element developed from Timoshenko beam theory, such as the DSG element, the shear correction factor is one of the necessary terms. This factor obtained from the shear strain energy from equilibrium equations equated to the shear strain energy obtained from the constitutive equation (Meena et al., 2012). In isotropic rectangular beams, this factor is equal to k = 5/6. For many applications of FGMs in beam structures, the shear correction factor is assumed to be constant. This factor is not constant for non-isotropic material like FGMs. This factor has a significant effect when analyzing thick beam problems in which the shear energy is crucial, which many papers have dealt with. (Timoshenko, 1922; Nguyen et al., 2008; Hosseini-Hashem et al. 2010).

 This paper aims to study the effect of the shear correction factor on the FGMs beam by using the DSG element. The equation for the shear correction factor is taken from Meena et al. (2012). FGMs beams with various boundary conditions and length thickness ratios (L/h = 4) are evaluated. Different power-law indexes (n) are also assessed to understand the convergence behavior of the DSG element in the FGMs beam.

    From the analysis, it can be concluded that the performance of DSG elements on FGMs beams yields good results.  The convergence results for k = 5/6 and k FGMs are close to the HOSDT reference for the number of elements ? 16. The top and bottom elastic modulus (Et/Eb) ratio and the power-law index (n) influence the shear correction factor. The greater the Et/Eb ratio, the greater the effect of the shear correction factor. In the case with Et/Eb = 0.35 for thick beams (L/h = 4), the error of displacement ranged from 0.184% (clamped-free) to 1.394% (clamped-clamped). In the case with Et/Eb = 20 for thick beams (L/h = 4), the error of displacement ranged from 4.432% (clamped-free) to 25.751%. (clamped-clamped). The largest error occurred in the clamped-clamped support shear energy is more important here than in other supports.

 The financial support from Penelitian Dasar Unggulan Perguruan Tinggi 2021 Nomor: NKB-199/UN2.RST/HKP.05.00/2021 is gratefully acknowledged.

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